Schwarz–Christoffel Mapping from a Rectangle

We investigate the conformal mapping of a rectangle onto a polygon. Using theta elliptic functions, we represent such a mapping via an integral of the Schwarz–Christoffel type; this representation is convenient for finding the conformal modulus of a quadrilateral or a generalized quadrilateral. We also write the Schwarz–Christoffel formula for a mapping from the triangle with angles , , . We present several examples of mapping from a rectangle. Besides, we give an implicit formula for the conformal modulus of a quadrilateral.